Probability density function as number of particles per unit volume? In this book Quantum Mechanics by P.J.E. Peebles pg 365 it hints at the idea of the wave function been the probability of  finding $n$ particles per unit volume. I have looked in other books and on websites, but can't find any more on this view. So in general can we relate the wave function to the number of particles per unit volume and if so how and if not why not?
 A: Imagine you stuck a detector in front if the target and it went off a certain number of times per second (on average) per square meter of cross section.
If you then placed it somewhere else at an angle from the forward direction it (the detector) would also go off a certain number of times per second (on average) per square meter of cross section.
Super. That's what you actually measure in the label. A bigger detector and more time gives (on average) more detection events.
A faster beam would make detection events more frequently. But you can normalize that. If it travels at speed v and you have cross section A and you get m detection events in time t then you can consider the volume of volume A(vt) and its highly similar to it having m particles in it. So m detection events in time t with cross section A with particles at speed v gives a relationship like A(vt)n=m where you can think of n as a number of particles per volume.
That's nice, not because there are n particles in some volume somewhere but because you can easily relate n to experimental parameters.
And obviously the wavefunction isn't literally a probability per unit volume (it is complex). But you are use to thinking of $|\psi(x,y,z,t)|^2$ as a probability per unit volume (it isn't). So it's an easier thing to compare with when you compare two things that are probabilities per unit volume.
In reality what you observe is detection events happening at different frequencies. That is always what you measure. Even when you consider $|\psi(x,y,z,t)|^2$ as a probability per unit volume when you look at the experimental data, it is different frequencies of detection events that you end up seeing in the data.
